use properties to rewrite the given equation. which equations have the same solution as the equation\n\\(…

use properties to rewrite the given equation. which equations have the same solution as the equation\n\\( \\frac { 3 } { 5 } x + \\frac { 2 } { 3 } + x = \\frac { 1 } { 2 } - \\frac { 1 } { 5 } x \\)\nselect three options.\n\\( \\square \\frac { 8 } { 5 } x + \\frac { 2 } { 3 } = \\frac { 1 } { 2 } - \\frac { 1 } { 5 } x \\)\n\\( \\square 18 x + 20 + 30 x = 15 - 6 x \\)\n\\( \\square 18 x + 20 + x = 15 - 6 x \\)\n\\( \\square 24 x + 30 x = - 5 \\)\n\\( \\square 12 x + 30 x = - 5 \\)

use properties to rewrite the given equation. which equations have the same solution as the equation\n\\( \\frac { 3 } { 5 } x + \\frac { 2 } { 3 } + x = \\frac { 1 } { 2 } - \\frac { 1 } { 5 } x \\)\nselect three options.\n\\( \\square \\frac { 8 } { 5 } x + \\frac { 2 } { 3 } = \\frac { 1 } { 2 } - \\frac { 1 } { 5 } x \\)\n\\( \\square 18 x + 20 + 30 x = 15 - 6 x \\)\n\\( \\square 18 x + 20 + x = 15 - 6 x \\)\n\\( \\square 24 x + 30 x = - 5 \\)\n\\( \\square 12 x + 30 x = - 5 \\)

Answer

Answer:

  • $\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$
  • $18x + 20+30x=15 - 6x$
  • $24x+30x=-5$

Explanation:

Step1: Combine like - terms for the first option

Combine $\frac{3}{5}x$ and $x$ ($x=\frac{5}{5}x$) in the original equation $\frac{3}{5}x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$. $\frac{3}{5}x+\frac{5}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$, so $\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$.

Step2: Eliminate fractions for the second option

Multiply the original equation $\frac{3}{5}x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$ by $30$ (the LCM of $5$, $3$, and $2$). $30\times(\frac{3}{5}x)+30\times\frac{2}{3}+30\times x = 30\times\frac{1}{2}-30\times\frac{1}{5}x$. $18x + 20+30x=15 - 6x$.

Step3: Simplify the second - obtained equation for the third option

Combine like - terms in $18x + 20+30x=15 - 6x$. $(18x+30x)+20=15 - 6x$. $48x+20 = 15-6x$. Add $6x$ to both sides and subtract $20$ from both sides: $48x+6x=15 - 20$. $24x+30x=-5$ (since $48x + 6x=(24 + 30)x$).